Adaptive fractal image compression using PSO

Available online at www.sciencedirect.com Procedia Computer Science 00 (2010) 000–000 Procedia Computer Science 2 (2010) 338–344 Procedia Computer Science www.elsevier.com/locate/procedia www.elsevier.com/locate/procedia ICEBT 2010 Adaptive Fractal Image Compression using PSO A.Muruganandhama, *, Dr. R.S.D. Wahida Banub a Sona College of Technology, Salem, India b Govt Engineering College, Salem, India Abstract At low bitrate and with acceptable quality in Fractal Image Compression (FIC) of the coded image, the encoding time is very large for most existing algorithms. In this paper, a fast fractal encoding system is proposed using particle swarm optimization (PSO) to reduce the encoding time. Here, an optimization technique is used for the MSE based on the stopping criterion between range block and domain block. This PSO technique can speedup the fractal encoder and preserve the image quality for medical imaging. c 2010 Published by Elsevier Ltd Open access under CC BY-NC-ND license. ⃝ Keywords- Fractal Image Compression (FIC);Mean Square Error (MSE) ;Particle Swarm Optimization (PSO); Iteration Function System (IFS). 1. Introduction The idea of the image redundancies can be efficiently exploited by means of, block self-affine transformations may call the fractal image compression (FIC). The first practical fractal image compression scheme was introduced in 1992 by Jacquin [3, 4]. The fractal transform for image compression was introduced in 1985 by Barnsley and Demko [1, 2]. The very high encoding time is the main disadvantages because of exhaustive search strategy. Therefore, decreasing the encoding time is an interesting research topic for FIC [3, 4]. One way of decreasing the encoding time is by using stochastic optimization methods using Genetic Algorithm (GA) this recent topics of GA-based methods are proposed to improve the efficiency [5, 6]. The idea of special correlation of an image is used in these methods, while the chromosomes in GA consist of all range blocks which leads to high encoding speed. Other researchers focused on improvements by tree structure search methods [12, 13]of the search process and parallel search methods [14, 15] or quad tree partitioning of range blocks [9,16] to make it faster. In this paper present a fast fractal encoding using particle swarm optimization. The outline of the remaining part of this paper is as follows: Section II includes fractal image coding. Section III involves implementation of PSO search. Section IV concerns the proposed fast fractal encoder using PSO optimization, and in Section V experimental results and comparisons are included. In Section VI present our conclusions. 2. Fractal Image Compression Algorithm The fundamental idea of fractal image compression is based on an Iteration Function System (IFS) in which the governing theorems are the Collage Theorem and the Contractive Mapping Fixed-Point Theorem [7]. The encoding unit of FIC for given gray level image of size N x N is (N/L)2 of non-overlapping range blocks of size L x L which forms the range pool R. For each range block v in R, one search in the (N - 2L + 1)2 overlapping domain blocks of size 2L x 2L which forms the domain pool D to find the best match. The parameters describing this fractal affine transformation of domain block into range block form the fractal compression code of v. * Corresponding Author. Tel.:+91-9443825711 E-mail address: muruga_salem@rediffmail.com c 2010 Published by Elsevier Ltd Open access under CC BY-NC-ND license. 1877-0509 ⃝ doi:10.1016/j.procs.2010.11.044 www.Matlabi.ir www.Matlabi.ir 339 A. Muruganandham, R.S.D. Wahida Banu / Procedia Computer Science 2 (2010) 338–344 The parameters of fractal affine transformation is Φ of domain block into range block having domain block coordinates ( tx, ty ), Dihedral transformation-d, contrast scaling-p, brightness offset-q. The affine transformation is illustrated as flowchart in fig. 1 for this fractal. ⎡ a 11 ⎤  = ⎢a ⎢ 21 ⎢u (x, y ) ⎦ ⎣ ⎣ 0 ⎡ φ ⎢ y a12 ⎤ a 22 ⎦ ⎡ a11 ⎢a Where the 2 x 2 sub-matrix ⎣ 21 ⎡1 0⎤ T0 = ⎢⎣0 1 ⎦, x y ⎤ ,  ⎦ (1) is one of the Dihedral transformations in (2) ⎡1 0 ⎤ T1 = ⎢⎣0 − 1⎦, ⎡0 − 1⎤ T6 = ⎢⎣1 0 ⎦, ⎡tx ⎤  + ⎢t y ⎢  ⎢  p ⎦ ⎣u ( x, y )⎦ ⎣ q 0 ⎤ ⎡ 0  ⎢ a 12 a 22 0 x ⎡− 1 0⎤ ⎡− 1 0 ⎤ ⎡0 1 ⎤ T2 = ⎢⎣ 0 1⎦ , T3 = ⎢⎣ 0 − 1⎦, T4 = ⎢⎣1 0⎦, ⎡ 0 1⎤ T5 = ⎢⎣− 1 0⎦, ⎡ 0 − 1⎤ T7 = ⎢⎣− 1 0 ⎦ (2) The above parameters are found using the following procedure 1. the domain block is first down-sampled to L x L and denoted by u 2. The down-sampled block is transformed subject to the eight transformations Tk: k = 0,. . . ,7 in the Dihedral on the pixel positions and are denoted by uk, k = 0,1, . . . ,7, where u0 = u. The transformations T1 and T2 correspond to the flips of u along the horizontal and vertical lines, respectively. T3 is the flip along both the horizontal and vertical lines. T4, T5, T6, and T7 are the transformations of T0, T1, T2, and T3 performed by an additional flip along the main diagonal line, respectively. 3. For each domain block, there are eight separate MSE computations required to find the index d such that d = arg min{MSE(( p k u k + q k ), v) : k = 0,1,...,7} MSE (u , v) = Where 1 L2 (3) L −1 ∑ (u(i, j = 0) − v(i, j )) 2 (4) i , j =0 Here, pk and qk can be computed directly as pk = qk = [L 2 uk ,v − ∑ ∑ L −1 L −1 i=0 j=0 ⎡ L2 u , u − k k ⎢⎣ 1 L2 [∑ L −1 i =0 ∑ L −1 j =0 (∑ u k (i, j ) ∑ i = 0 ∑ L −1 L −1 i=0 ∑ L −1 j=0 L −1 j=0 ) ] v (i, j ) − p k ∑i =0 ∑ j = 0 u k (i, j ) . L −1 v (i, j ) u k (i, j ) ⎤ ⎦ 2 L −1 ], (5) (6) www.Matlabi.ir www.Matlabi.ir 340 A. Muruganandham, R.S.D. Wahida Banu / Procedia Computer Science 2 (2010) 338–344 Input image of size N x N Partition input image into (N/L)2 non overlapping range blocks of size L x L and (N2L+1)2 overlapping domain blocks of size 2L x 2L For Range blocks 1 to (N/L) 2 For Domain blocks 1 to (N-2L+1)2 For Orientation 1 to 8 for increasing the speed Calculate Mean Square Error Scaling, and Offset Store quantized scaling and offset, orientation and position of the domain block of minimum MSE Fig. 1 Fractal affine transformation of domain block into range block 4. As u runs over all of the domain blocks in D to find the best match, the terms tx and ty can be obtained together with d and the specific p and q corresponding this d, the affine transformation (1) is found for the given range block v. In practice, tx, ty, d, p, and q can be encoded using log2 (N), log2 (N), 3, 5, and 7 bits, respectively, which are regarded as the compression code of v. Finally, the encoding process is completed as v runs over all of the (N/L) 2 range blocks in R. 16 scale bits = 3 4 5 6 7 8 9 10 11 14 MSE 12 10 8 6 4 3 4 5 6 7 8 9 no. of bits for Offset quantization 10 11 Fig. 2 MSE Vs. quantization parameter From fig. 2 find the MSE vs. quantization parameter for randomly selected range block of size 8 x 8 from 256 x 256 Lena image. From fig. 2, choosing 5 bits and 7 bits as quantization parameter for scale and offset value are justified respectively. To decode, the compression codes to obtain a new image, and proceeds recursively by chooses any image as the initial one and makes up the (N/L) 2 affine transformations. According to Partitioned Iteration Function Theorem (PIFS), the sequence of images will converge. According to user’s application the stopping criterion of the recursion is designed based on the PSO. The final image is the retrieved image of fractal coding. 3. Particle Swarm Optimization (PSO) A population-based algorithm is PSO for searching global optimum. To simulate a simplified social behaviour [8, 9] is the way of original idea of PSO. Similar to the crossover operation of the GA, in PSO the particles are adjusted toward the best individual experience (PBEST) and the best social or global experience (GBEST). However, PSO is unlike a GA, why because in www.Matlabi.ir www.Matlabi.ir 341 A. Muruganandham, R.S.D. Wahida Banu / Procedia Computer Science 2 (2010) 338–344 that each potential solution, particle is ‘‘flying” through hyperspace with a velocity, the particles and the swarm have memory for process; in the population of the GA memory does not exist. Let xj,d(t) and vj,d(t) denote the dth dimensional value of the vector of position and velocity of jth particle in the swarm, respectively, at time t. The PSO model can be expressed as x v j ,d (t ) = v j ,d (t − 1) + c1 .ϕ1 .( x *j ,d − x j ,d (t − 1)) + c 2 .ϕ 2 .( xd# − x j ,d (t − 1)), (7) x j ,d (t ) = x j ,d (t − 1) + v j ,d (t ), (8) * j ,d x # d where (PBEST) denotes the best position of jth particle up to time t-1 and (GBEST) denotes the best position of the whole swarm up to time t-1, φ1 and φ2 are random numbers, and c1 and c2 represent the individuality and sociality coefficients, respectively. Input image of size N x N (Start) Partition input image into (N/L)2 non overlapping range blocks of size L x L and (N-2L+1)2 overlapping domain blocks of size 2L x 2L Initialization Fitness evaluation For Range blocks 1 to (N/L) 2 Experience updating Calculate scale, offset, orientation and position of the domain block of minimum MSE from all domain blocks using PSO Moving No Stopping criterion reached? Store quantized scaling and offset, orientation and position of the domain block of minimum MSE Yes (Stop) Fig. 3 Block diagram of PSO Fig.4 Fractal encoding of proposed method The steps involved here is the population size is first determined, and the velocity and position of each particle are initialized. Each particle moves according to (7) and (8), and the fitness is then calculated. Meanwhile, the best positions of each swarm and particles are recorded. Finally, as the stopping criterion is satisfied, the best position of the swarm is the final solution. The block diagram of PSO is displayed in Fig. 3 and the main steps are given as follows: 1. Set the swarm size. Initialize the velocity and the position of each particle randomly. 2. For each j, evaluate the fitness value of xj and update the individual best position 3. Find the new best position of the whole swarm. Update the swarm best position x# if the fitness of the new best position is better than that of the previous swarm. 4. If the stopping criterion is satisfied, then stop. 5. For each particle, update the position and the velocity according (8) and (7). Go to step 2. x *j ,d if better fitness is found. 4. Proposed Method In the proposed fast fractal encoding using PSO, reduce the encoding time by reducing the searching time to find a best match domain block for the given range block from all domain blocks. Flowchart of the fractal encoding of the proposed method is shown in fig. 4. Domain block of minimum MSE and stopping criterion is found by PSO using the steps given below: 1. Set the swarm size must be proportional to (N-2L+1)2/(maximum no. of iterations for PSO) and initialize the each particle velocity and the position randomly www.Matlabi.ir www.Matlabi.ir 342 A. Muruganandham, R.S.D. Wahida Banu / Procedia Computer Science 2 (2010) 338–344 2. By using eqn. (3) the fitness value includes finding , MSE between domain block specified by the particles position and given range block 3. Update swarm best position if the fitness of the new best position is better than that of the previous swarm. 4. If swarm best position is not changed for some percentage of maximum iteration for PSO, then stop 5. The best position of the particles is updated using eqn. (7) and (8) and go to step 2. 5. Experimental Results The results have been compared to the full search FIC mentioned in the previous sections in terms of encoding time and PSNR of fast fractal encoding using PSO Figures of 5a, 5b, 5c shows the original images Lena, Goldhill, Cameraman 256 x 256 at 0.5bpp and Figures of 6a, 6b, 6c shows the decoded Lena, Goldhill, Cameraman image using full search FIC and fast fractal encoding using PSO. Fig. 5a.Original 256 x 256 Lena Image (a) (b) Fig. 6. Decoded Lena image at 0.5 bits per pixel for (a) Full search FIC (b) proposed fast fractal encoding using PSO Figure 9c shows the variation in PSNR by varying the stopping criterion in fast fractal encoding using PSO by changing the percentage of maximum iteration of PSO. From figure 9c the variation of PSNR with variation of stopping criterion is very less. Hence a 10% of maximum iteration for PSO is chooses as stopping criterion with example as shown from the figure (9a, 9b, 9d).for various swarm size. (a) Fig. 5b. Original 256 x 256 Goldhill Image Fig. 5c. Original 256 x 256 Camera man Image (b) Fig. 7. Decoded Goldhill image at 0.5 bits per pixel for (a) Full search FIC b) proposed fast fractal encoding using PSO (a) (b) Fig. 8. Decoded Camera man image at 0.5 bits per pixel for (a) Full search FIC (b) proposed fast fractal encoding using PSO www.Matlabi.ir www.Matlabi.ir 343 A. Muruganandham, R.S.D. Wahida Banu / Procedia Computer Science 2 (2010) 338–344 98 106 nop=20 nop=24 nop=28 nop=32 105 104 103 97 96 95 nop=20 102 94 MSE 93 MSE 101 100 nop=24 nop=28 nop=32 92 99 91 98 90 97 96 0.1 0.15 0.2 0.25 0.3 0.35 stopping criterion 0.4 0.45 0.5 89 0 0.02 0.04 0.06 0.08 0.1 012 0.14 0.16 0.18 0.2 stopping criterion Figure 9b. MSE vs. Stopping criterion for camera man512 x 512 Figure 9a. MSE vs. Stopping criterion for Lena image512 x 512 106 nop=20 34.34 105 34.338 104 34.336 103 nop=24 nop=28 nop=32 102 34.334 PSNR (dB) MSE 101 34.332 100 34.33 99 34.328 98 34.326 97 34.324 10 20 30 40 50 60 70 Stopping criterion (percentage of maximum iteration for PSO) 80 Figure 9c. variation in PSNR by varying the stopping criterion 96 0.1 0.15 0.2 0.25 0.3 0.35 stopping criterion 0.4 0.45 0.5 Figure 9d. MSE vs. Stopping criterion for Goldhill512 x 512 The numeric results containing bitrates, encoding time and PSNR of decoded image of various images are tabulated in table I. TABLE I comparison results of encoding time and PSNR of Lena, Goldhill, Camera man images Input image Bitrates (bpp) Lena 0.5 Goldhill 0.5 Camera man 0.5 Method Full search FIC Proposed method Full search FIC Proposed method Full search FIC Proposed method Encoding Time (hh: mm: ss) 09:07:20 00:15:34 09:02:12 00:17:37 09:02:49 00:15:24 PSNR (dB) 35.80 35.03 33.64 32.77 35.11 34.23 6. Conclusion Fractal image compression can produce better compression ratio at acceptable quality. By using PSO for fractal coding can reduce the encoding time with 1.2db loss in image quality. This can be improved by applying some classification technique and various FIC methods. Due to the fast encoding reasons this may be used in medical imaging (Through Internet Data Transfer) www.Matlabi.ir www.Matlabi.ir 344 A. Muruganandham, R.S.D. Wahida Banu / Procedia Computer Science 2 (2010) 338–344 References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] M.F. Barnsley, S. Demko, Iterated function systems and the global construction of fractals, Proc. Roy. Soc. Lond A399 (1985) 243–275. A.E. Jacquin, Fractal image coding: a review, Proc. IEEE 10 (1993)1451–1465. A.E. Jacquin, Image coding based on a fractal theory of iterated contractive image transformations, IEEE Transactions on Signal Processing 1 (1992) 18– 30. M.F. Barnsley, A.D. 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